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31
Unified analysis of discontinuous Galerkin methods for elliptic problems
 SIAM J. Numer. Anal
, 2001
"... Abstract. We provide a framework for the analysis of a large class of discontinuous methods for secondorder elliptic problems. It allows for the understanding and comparison of most of the discontinuous Galerkin methods that have been proposed over the past three decades for the numerical treatment ..."
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Cited by 525 (31 self)
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Abstract. We provide a framework for the analysis of a large class of discontinuous methods for secondorder elliptic problems. It allows for the understanding and comparison of most of the discontinuous Galerkin methods that have been proposed over the past three decades for the numerical treatment of elliptic problems.
An A Priori Error Analysis Of The Local Discontinuous Galerkin Method For Elliptic Problems
, 2000
"... . In this paper, we present the first a priori error analysis for the Local Discontinuous Galerkin method for a model elliptic problem. For arbitrary meshes with hanging nodes and elements of various shapes, we show that, for stabilization parameters of order one, the L 2 norm of the gradient and ..."
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Cited by 96 (25 self)
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. In this paper, we present the first a priori error analysis for the Local Discontinuous Galerkin method for a model elliptic problem. For arbitrary meshes with hanging nodes and elements of various shapes, we show that, for stabilization parameters of order one, the L 2 norm of the gradient and the L 2 norm of the potential are of order k and k + 1=2, respectively, when polynomials of total degree at least k are used; if stabilization parameters of order h \Gamma1 are taken, the order of convergence of the potential increases to k + 1. The optimality of these theoretical results are tested in a series of numerical experiments on two dimensional domains. Key words. Finite elements, discontinuous Galerkin methods, elliptic problems AMS subject classifications. 65N30 1. Introduction. In this paper, we present the first a priori error analysis of the Local Discontinuous Galerkin (LDG) method for the following classical model elliptic problem: \Gamma\Deltau = f in\Omega ; u ...
Local Discontinuous Galerkin Methods For The Stokes System
, 2000
"... In this paper, we introduce and analyze local discontinuous Galerkin methods for the Stokes system. For arbitrary meshes with hanging nodes and elements of various shapes we derive a priori estimates for the L²norm of the errors in the velocities and the pressure. We show that optimal order estimat ..."
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Cited by 45 (17 self)
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In this paper, we introduce and analyze local discontinuous Galerkin methods for the Stokes system. For arbitrary meshes with hanging nodes and elements of various shapes we derive a priori estimates for the L²norm of the errors in the velocities and the pressure. We show that optimal order estimates are obtained when polynomials of degree k are used for each component of the velocity and polynomials of degree k  1 for the pressure, for any k >= 1. We also consider the case in which all the unknowns are approximated with polynomials of degree k and show that, although the orders of convergence remain the same, the method is more efficient. Numerical experiments verifying these facts are displayed.
Stabilization mechanisms in discontinuous Galerkin finite element methods
 Comput. Methods Appl. Mech. Engrg
, 2006
"... In this paper we propose a new general framework for the construction and the analysis of Discontinuous Galerkin (DG) methods which reveals a basic mechanism, responsible for certain distinctive stability properties of DG methods. We show that this mechanism is common to apparently unrelated stabili ..."
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Cited by 44 (6 self)
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In this paper we propose a new general framework for the construction and the analysis of Discontinuous Galerkin (DG) methods which reveals a basic mechanism, responsible for certain distinctive stability properties of DG methods. We show that this mechanism is common to apparently unrelated stabilizations, including jump penalty, upwinding, and Hughes–Franca type residualbased stabilizations.
WellPosedness For NonIsotropic Degenerate ParabolicHyperbolic Equations
 ANN. INST. H. POINCARÉ ANAL. NON LINÉAIRE
, 2003
"... We develop a wellposedness theory for solutions in L1 to the Cauchy problem of general degenerate parabolichyperbolic equations with nonisotropic nonlinearity. A new notion of entropy and kinetic solutions and a corresponding kinetic formulation are developed which extends the hyperbolic case. Th ..."
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Cited by 41 (6 self)
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We develop a wellposedness theory for solutions in L1 to the Cauchy problem of general degenerate parabolichyperbolic equations with nonisotropic nonlinearity. A new notion of entropy and kinetic solutions and a corresponding kinetic formulation are developed which extends the hyperbolic case. The notion of kinetic solutions applies to more general situations than that of entropy solutions; and its advantage is that the kinetic equations in the kinetic formulation are well defined even when the macroscopic fluxes are not locally integrable, so that L1 is a natural space on which the kinetic solutions are posed. Based on this notion, we develop a new, simpler, more effective approach to prove the contraction property of kinetic solutions in L1, especially including entropy solutions. It includes a new ingredient, a chain rule type condition, which makes it different from the isotropic case.
Optimal a priori error estimates for the hpversion of the local discontinuous Galerkin method for convectiondiffusion problems
 Math. Comp
"... Abstract. We study the convergence properties of the hpversion of the local discontinuous Galerkin finite element method for convectiondiffusion problems; we consider a model problem in a onedimensional space domain. We allow arbitrary meshes and polynomial degree distributions and obtain upper b ..."
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Cited by 35 (7 self)
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Abstract. We study the convergence properties of the hpversion of the local discontinuous Galerkin finite element method for convectiondiffusion problems; we consider a model problem in a onedimensional space domain. We allow arbitrary meshes and polynomial degree distributions and obtain upper bounds for the energy norm of the error which are explicit in the meshwidth h, in the polynomial degree p, and in the regularity of the exact solution. We identify a special numerical flux for which the estimates are optimal in both h and p. The theoretical results are confirmed in a series of numerical examples. 1.
The Local Discontinuous Galerkin Method for Contaminant Transport
"... We develop a discontinuous finite element method for advectiondiffusion equations arising in contaminant transport problems, based on the Local Discontinuous Galerkin method of Cockburn and Shu [14]. This method is defined locally over each element, thus allowing for the use of different approximat ..."
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Cited by 28 (10 self)
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We develop a discontinuous finite element method for advectiondiffusion equations arising in contaminant transport problems, based on the Local Discontinuous Galerkin method of Cockburn and Shu [14]. This method is defined locally over each element, thus allowing for the use of different approximating polynomials in different elements, furthermore the elements do not have to conform, or "matchup" at interfaces. The method has a builtin upwinding mechanism for added stability. Moreover, it is conservative. We describe the method for multidimensional systems of equations with possibly nonlinear adsorption terms, and provide some numerical results in both one and two dimensions. These results examine the accuracy of the method, and its ability to approximate solutions to some linear and nonlinear problems arising in contaminant transport.
The hplocal discontinuous Galerkin method for lowfrequency timeharmonic Maxwell equations
 MATH. COMP
, 2001
"... The local discontinuous Galerkin method for the numerical approximation of the timeharmonic Maxwell equations in a lowfrequency regime is introduced and analyzed. Topologically nontrivial domains and heterogeneous media are considered, containing both conducting and insulating materials. The prese ..."
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Cited by 23 (9 self)
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The local discontinuous Galerkin method for the numerical approximation of the timeharmonic Maxwell equations in a lowfrequency regime is introduced and analyzed. Topologically nontrivial domains and heterogeneous media are considered, containing both conducting and insulating materials. The presented method involves discontinuous Galerkin discretizations of the curlcurl and graddiv operators, derived by introducing suitable auxiliary variables and socalled numerical fluxes. An hpanalysis is carried out and error estimates that are optimal in the meshsize h and slightly suboptimal in the approximation degree p are obtained.
An analysis of the minimal dissipation local discontinuous Galerkin method for convectiondiffusion problems
 HYBRIDIZATION OF DG, MIXED AND CG METHODS 37
, 2006
"... Abstract. We analyze the socalled the minimal dissipation local discontinuous Galerkin method for convectiondiffusion or diffusion problems. The distinctive feature of this method is that the stabilization parameters associated with the numerical trace of the flux are identically equal to zero in ..."
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Cited by 20 (6 self)
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Abstract. We analyze the socalled the minimal dissipation local discontinuous Galerkin method for convectiondiffusion or diffusion problems. The distinctive feature of this method is that the stabilization parameters associated with the numerical trace of the flux are identically equal to zero in the interior of the domain; this is why its dissipation is said to be minimal. We show that the orders of convergence of the approximations for the potential and the flux using polynomials of degree k are the same as those of all known discontinuous Galerkin methods for both unknowns, namely, (k + 1) and k, respectively. Our numerical results verify that these orders of convergence are sharp. The novelty of the analysis is that it bypasses a seemingly indispensable condition, namely, the positivity of the above mentioned stabilization parameters, by using a new, carefully defined projection tailored to the very definition of the numerical traces. 1.